# Qualitative possibility theory

Qualitative possibility theory presupposes an ordinal representation of uncertainty in the form of a plausibility relation between possible worlds, that can be projected on a totally ordered qualitative scale. Possibilistic logic is then an extension of classical logic to knowledge bases where formulas are ordered according to their levels of certainty modelled by necessity measures. On top of possibility and necessity degrees induced by the plausibility ordering, two other set-functions can be defined, representing actual or guaranteed possibility and potential necessity.

- Belief revision : The AGM approach to theory revision is central in Artificial Intelligence and it implies a representation of priorities between formulas in agreement with possibility theory.
- Spohn’s rank functions encode possibility measures, under a specific format (by means of natural integers). This approach comes down to a system of infinitesimal probabilities.
- Non-monotonic reasoning : the inconsistency-tolerant inference relation in possibilistic logic is actually characterised by the properties of non-monotonic inference relations after Gabbay, Lehmann and Makinson.

More recently formal links between qualitative possibility theory and other topics in information processing and knowledge representation have been laid bare:

- Links with Formal Concept Analysis : The four set functions at work in possibility theory have counterparts in terms of Galois-like connections instrumental in FCA, which leads to new developments of FCA.
- Links with modal logic : The all-or-nothing version of possibility theory is exactly captured by a fragment of the modal logic KD called MEL, whose semantics in in terms of simple epistemic states. On this basis, a conjoint generalisation of possibilistic logic and MEL called generalized possibilistic logic has been devised. It yields an expressive multimodal logic that can encode answer-set programming and the equilibrium logic of Pierce, and logics of all I know. It enables to clarify the semantic content of formulas in such languages.
- Links with three-valued logics of incomplete information (noticeably Kleene logic) where the third truth-value refers to the idea of unknown, and some paraconsistent three-valued logics (noticeably Priest logic) where the third truth-value refers to the idea of contradiction. We could show that all such logics are captured by the MEL logic.
- Links with Belnap logic of contradiction: in fact it can be captured by monotonic all-or-nothing set functions and generalised by means of qualitative fuzzy measures measuring support in favor and against propositions.
- Links with the square of oppositions, a fundamental pattern of Aristotle logic.

In the long range, we aim at building a qualitative framework for uncertainty and qualitative information fusion, general enough to articulate qualitative extensions of possibility theory (qualitative capacities), some classes of non-regular modal logics, and some inconsistency-tolerant logics such as Belnap logic.

## References

- Zina Ait-Yakoub, Yassine Djouadi, Didier Dubois, Henri Prade.
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